\[ \boxed { {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) -{a}^{2} \left ( \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{5}+2\, \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{3}+{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 10.857379 (sec), leaf count = 35 \[ \text {DSolve}\left [y^{(3)}(x)-a^2 \left (y'(x)^5+2 y'(x)^3+y'(x)\right )=0,y(x),x\right ] \]
Maple: cpu = 0.265 (sec), leaf count = 95 \[ \left \{ y \left ( x \right ) =\int \!{\it RootOf} \left ( -3\,\int ^{{ \it \_Z}}\!{\frac {1}{\sqrt {3\,{{\it \_f}}^{6}{a}^{2}+9\,{{\it \_f}}^ {4}{a}^{2}+9\,{{\it \_f}}^{2}{a}^{2}+9\,{\it \_C1}}}}{d{\it \_f}}+x+{ \it \_C2} \right ) \,{\rm d}x+{\it \_C3},y \left ( x \right ) =\int \!{ \it RootOf} \left ( 3\,\int ^{{\it \_Z}}\!{\frac {1}{\sqrt {3\,{{\it \_f}}^{6}{a}^{2}+9\,{{\it \_f}}^{4}{a}^{2}+9\,{{\it \_f}}^{2}{a}^{2}+9 \,{\it \_C1}}}}{d{\it \_f}}+x+{\it \_C2} \right ) \,{\rm d}x+{\it \_C3} \right \} \]